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The book should not be used with young pupils; it is essential that the various topics should be first introduced as in Practical Geometry or Junior Geometry. Special points in this book are:1. The various topics are arranged in chapters which are more

or less self-contained. 2. There is a large supply of riders, which are carefully graded

and arranged in groups, generally at the ends of the chapters.

Hints are supplied in some cases. 3. The treatment is Euclidean throughout, with a modern

sequence. Practically no numerical work is introduced, that

being left to Practical Geometry. 4. Separate proofs are given of a few corollaries where their

importance seems to demand it. 5. At the end of the book there is a chapter introducing the

pupil to more advanced Geometry, including some riders on

Solid Geometry.
6. There is a set of revision papers.

7. There is a set of hard Miscellaneous Exercises. We have to acknowledge the courtesy of H.M. Stationery Office, of the Oxford and Cambridge Joint Board, of the Oxford Local Examinations Delegacy, of the Cambridge Local Examinations Syndicate, of the Universities of London and Bristol, and of the Board for the Common Examination for Entrance to Public Schools for permission to include exercises from public examination papers.

Finally we have to thank many colleagues and friends for suggestions and criticisms.

A. W. S.

R. T. H. HARROW,

May, 1926.

TABLE OF CONTENTS

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Chapter II.

ANGLES AT A POINT.

ITHEOREM 1. If a straight line stands on another straight line, the

sum of the two angles so formed is equal to two right angles .

CoR. If any number of straight lines meet at a point, the sum of

all the angles made by consecutive lines is equal to four right

angles

THEOREM 2. If the sum of two adjacent angles is equal to two right

angles, the exterior arms of the angles are in the same straight

line

NOTE ON A THEOREM AND ITS CONVERSE

ITHEOREM 3. If two straight lines intersect, the vertically opposite

angles are equal

PARALLEL STRAIGHT LINES.

$THEOREM 4. When a straight line cuts two other straight lines, if

(i) a pair of alternate angles are equal,

or (ii) a pair of corresponding angles are equal,

or (iii) a pair of interior angles on the same side of the cutting

line are together equal to two right angles,

then the two straight lines are parallel

Cor. If each of two straight lines is perpendicular to a third

straight line, the two straight lines are parallel to one another

PLAYFAIR'S AXIOM .

ITHEOREM 5. If a straight line cuts two parallel straight lines,

(i) alternate angles are equal,

(ii) corresponding angles are equal,

(iii) the interior angles on the same side of the cutting line

are together equal to two right angles .

{THEOREM 6. Straight lines which are parallel to the same straight

line are parallel to one another

ITAEOREM 7. If straight lines are drawn from a point parallel to

the arms of an angle, the angle between those straight lines

is equal or supplementary to the given angle.

Some examining bodies do not ask for proofs of theorems marked I. In each
case the various examination schedules should be consulted.

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I Some examining bodies do not ask for proofs of theorems marked I. In each
case the various examination schedules should be consulted.

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36

39

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THEOREM 15. If two right-angled triangles have their hypotenuses

equal, and one side of the one equal to one side of the other,

the triangles are congruent

24

NOTE ON THE AMBIGUOUS CASE

24

EXERCISES

25

Chapter IV.

CONSTRUCTIONS

30

Chapter V.

INEQUALITIES.

THEOREM 16. If two sides of a triangle are unequal, the greater

side has the greater angle opposite to it .

THEOREM 17. If two angles of a triangle are unequal, the greater

angle has the greater side opposite to it .

36

THEOREM 18. Any two sides of a triangle are together greater

than the third side.

37

THEOREM 19. Of all the straight lines that can be drawn to a

given straight line from a given point outside it, the perpen-

dicular is the shortest

EXERCISES

39

Chapter VI.

NOTE ON DEFINITIONS

42

THE PARALLELOGRAM.

THEOREM 20. (i) The opposite sides of a parallelogram are equal . 43

(ii) The opposite angles of a parallelogram are equal 43

(iii) Each diagonal bisects the parallelogram

43

(iv) The diagonals of a parallelogram bisect one another . 44

Cor. 1. If a parallelogram has one of its angles a right angle, all

its angles must be right angles

44

COR. 2. If one pair of adjacent sides of a parallelogram are equal,

all its sides are equal

44

THEOREM 21. (i) A quadrilateral is a parallelogram if both pairs

of opposite angles are equal

44

(ii) A quadrilateral is a parallelogram if both pairs of

opposite sides are equal

45

(iii) A quadrilateral is a parallelogram if one pair of

opposite sides are equal and parallel .

45

(iv) A quadrilateral is a parallelogram if its diagonals

bisect one another

45

Cor. If equal perpendiculars are erected on the same side of a

straight line, the straight line joining their extremities is

parallel to the given line

45

I Some examining bodies do not ask for proofs of theorems marked I. In each

case the various examination schedules should be consulted.

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